Special Relativity - Doppler Effect

The ultraviolet Lyman alpha line of hydrogen with wavelength 121.5 nanometers is emitted by an astronomical object. An observer on earth measures the wavelength of the light received from the object to be 607.5 nanometers. The observer can conclude that the object is moving with radial velocity of

A. m/s toward Earth
B. m/s toward Earth
C. m/s away from Earth
D. m/s away from Earth
E. m/s away from Earth

(GR0177 #71)

Solution:

Given:
λ₀ = 121.5 nm
λ = 607.5 nm

λ  λ₀ → the object is moving away (receding)

(A) and (B) are FALSE.

(E) is FALSE since v is larger than c.

Doppler Effect for light:


with

and
+ sign = approaching
− sign = receding

Since the object and the source are receding:

λ = λ₀(1 + β)^1/2/(1 − β)^1/2
λ/λ₀ = (1 + β)^1/2/(1 − β)^1/2
607.5/121.5 = 5 = (1 + β)^1/2/(1 − β)^1/2
25(1 − β) = (1 + β)
24 = 26β
β = 24/26 = 12/13 = v_source/c
v_source = (12/13)c = (12/13)(3 × 10⁸) = (36/13) × 10⁸ = 2.76 × 10⁸ m/s

Answer: D